Okay, I've decided to end this specs is meaningless crap comment with facts. something m$ doesn't like to give out. So the cpu is practically identical. So that leaves the ram and gpu. I'll start first with the gpu.

x1 has a gpu compared to an ati 7790 same teraflops; while the ps4 has a gpu compared with an ati 7870, also same teraflops.-http://www.extremetech.com/gaming/156273-xbox-720-vs-ps4-vs-pc-how-the-hardware-specs-compare

In the review they say both should be similar, but as any pc gamer will tell you the gpu is what drives games. When mark cerny asked developers what they wanted; they said give us a powerful gpu. cpu is hardly ever a bottleneck and faster ram helps with loading screens faster and how much of the 3d world can be generated in real-time at once. 

As you can see from this review:http://www.guru3d.com/articles_pages/club3d_radeon_7870_joker_review,17.html of the 7870

The 7870 gains over the 7790 are massive. 46 fps to 29 in bf3. 30 to 21 in crysis 3. 50% gains like promised. note the only difference between the setups was the gpu. this isn't even taking into count the faster and more ram the ps4 has.

So yes, specs do matter and with the ps4 and x1 on the same architecture the gains will be similar. If nothing else games will run a LOT better on the ps4 and exclusives will look better too. The only exception is if developers are lazy and do bad coding, but I doubt this will happen. 

I'm writing this now to show people I was right when the ps4 does release later this year.